File:Rotating spherical harmonics.gif

Rotating_spherical_harmonics.gif(445 × 445 pixels, file size: 799 KB, MIME type: image/gif, looped, 21 frames)

Description
English: Rotating spherical harmonics, ${\displaystyle l=0}$ to 4 vertical, ${\displaystyle m=0}$ to ±4 horizontal
Deutsch: Veranschaulichung einiger Kugelflächenfunktionen (um die z-Achse rotierend). Dargestellt ist ${\displaystyle Y_{l,m}}$, wobei l der Zeile und m der Spalte entspricht. Zeilen und Spalten werden jeweils bei null beginnend durchnummeriert
Date
Source Own work
Author Cyp
Permission
(Reusing this file)
 Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.

Source in C:

```//GPL (as if anyone would want this anyway...)
#include <stdio.h>
#include <math.h>

#define PI 3.14159265358979323846264338327950288

//m>=0
double cfac(int l, int m, double z) {
int x;
int i;
double r, s, t;
r=log((2*l+1)*(.25/PI))*.5;
r-=log(2)*l;
i=(l+m+1)>>1;
for(x=2;x<=l-i;++x) r-=   log(x);
if(l-m<i) {
/*for(   ;x<=l-m;++x) r+= 0*log(x);*/ x=l-m+1;
for(   ;x<=i  ;++x) r-=.5*log(x);
} else {
/*for(   ;x<=i  ;++x) r-= 0*log(x);*/ x=i+1;
for(   ;x<=l-m;++x) r+=   log(x);
}
for(   ;x<=l+m;++x) r+=.5*log(x);
for(   ;x<=2*i;++x) r+=   log(x);

if(m) r+=.5*log(2);

r=exp(r)*((l^i)&1?-1:1)*(2*i-l-m?z:1);
s=-z*z;
t=r;
for(i=((l+m+1)>>1)+1;i<=l;++i) t+=r*=s*(2*i-1)*(2*i)*(l-i+1)/((double)(2*i-l-m-1)*(2*i-l-m)*i);
return(t);
}

double cval(int m, double x, double y) {
int i;
_Complex double d=x+1i*y, r;
for(r=1;m;m>>=1,d*=d) if(m&1) r*=d;
return(__real__ r);
}

void rend(unsigned char *c, int sx, int sy, int pit, int l, int m, double ang) {
int a, b;
double x, y, z, sa, ca, f, t, r;
sa=sin(ang); ca=cos(ang);
for(b=0;b<sy;++b) {
for(a=0;a<sx;++a) {
z=1-b*2/(sy-1.);
x=a*2/(sx-1.)-1;
t=1-x*x-z*z;
if(t<0) t=1e99; else {
y=-sqrt(t);
t=cos(.2)*z-sin(.2)*y;
y=sin(.2)*z+cos(.2)*y;
z=t;
t=ca*x+sa*y;
y=-sa*x+ca*y;
x=t;
f=cfac(l, m, z);
t=cval(m, x, y)*f;//printf("a%lf, %lf, %lf, %lf\n", t, x, y, z);
}
r=128+t*512; if(r<0) r=0; else if(r>255) r=255; c[a*3+b*pit  ]=(int)r|31;
r=128+t*256; if(r<0) r=0; else if(r>255) r=255; c[a*3+b*pit+1]=(int)r|15;
r=128+t*128; if(r<0) r=0; else if(r>255) r=255; c[a*3+b*pit+2]=(int)r|7;
}
}
}

unsigned char pic[256*256*3];

int main(void) {
FILE *f;
char fn[100];
int a, l, m;
for(a=0;a<256*256*3;++a) pic[a]=255;
for(a=0;a<100;++a) {
for(l=0;l<=4;++l) for(m=0;m<=l;++m) rend(pic+m*27*3+l*27*134*3, 26, 26, 134*3, l, m, PI*2/100*a);
//    for(l=0;l<=4;++l) for(m=0;m<=l;++m) rend(pic+m*51*3+l*51*256*3, 50, 50, 256*3, l, m, PI*2/100*a);
//    rend(pic+4*51*3+0*51*256*3, 50, 50, 256*3, 47, 1, PI*2/100*a);
//    rend(pic+4*51*3+0*51*256*3, 50, 50, 256*3, 47, 11, PI*2/100*a);
//    rend(pic, 256, 256, 256*3, 5, 2, PI*2/100*a);
sprintf(fn, "sph%04d.ppm", a);
f=fopen(fn, "wb");
fprintf(f, "P6\n%d %d\n255\n", 134, 134);
fwrite(pic, 134*134*3, 1, f);
fclose(f);
}
/*  int l, m, i;
double r;
for(l=0;l<=7;++l) for(m=0;m<=l;++m) {
printf("%d,%d: %lf", l, m, r=cfac(l, m));
for(i=((l+m+1)>>1)+1;i<=l;++i) printf(", %lf", r*=(double)(2*i-1)*(2*i)*(l-i+1)/((double)(2*i-l-m-1)*(2*i-l-m)*i));
printf("\n");
}*/
return(0);
}
```

Source in Mathematica:

```Export["Rotating spherical harmonics.gif",
Table[
GraphicsGrid[
Table[
SphericalPlot3D[
1,
{\[Theta], 0, \[Pi]},
{\[CapitalPhi], 0, 2 \[Pi]},
ColorFunction -> Function[
{x, y, z, \[Theta], \[Phi], r},
Blend[
{RGBColor["#1F2F57"], RGBColor["#FFCFA7"]},
2*(Re[
SphericalHarmonicY[l, m, \[Theta], \[Phi]]*
Exp[- Sign[m] I t]] + 0.25)]],
ColorFunctionScaling -> False,
Mesh -> False,
Boxed -> False,
Axes -> False,
PlotPoints -> 10,
MaxRecursion -> 2,
ViewPoint -> Front,
ImageSize -> {80, 80}
],
{l, 0, 5},
{m, 0, l}
]
],
{t, 0, 2 \[Pi], \[Pi]/10}
]
]
```

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current12:06, 3 December 2016445 × 445 (799 KB)ChemmixChanged range to ℓ = 0, …, 4 as in original file.
11:54, 3 December 2016533 × 533 (1.1 MB)ChemmixBetter quality version generated in Mathematica: <code> Export[][][][][][Theta], 0, \[Pi]}, {\[CapitalPhi], 0, 2 \[Pi]}, ColorFunction -> Function[][Theta], \[Phi], r}, Blend[]["#1F2F57"], RGBColor["#FFCFA7"]}, 2*(Re[][][Theta]...
21:39, 12 August 2005134 × 134 (161 KB)CypRotating spherical harmonics, l=0 to 4 vertical, m=0 to ±4 horizontal, {{GFDL}}
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